in \({\mathbb {R}}^N\), where \((-\varDelta )^s_p\) is the fractional p-Laplacian operator, with \(0<s<1<p<\infty \) and \(ps<N\), the nonlinearity \(f:{\mathbb {R}}^N\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a Carathéodory function and satisfies the Ambrosetti–Rabinowitz condition, \(V:{\mathbb {R}}^N\rightarrow {\mathbb {R}}^+\) is a potential function and \(g:{\mathbb {R}}^N\rightarrow {\mathbb {R}}\) is a perturbation term. We first establish Batsch–Wang type compact embedding theorem for the fractional Sobolev spaces. Then multiplicity results are obtained by using the Ekeland variational principle and the Mountain Pass theorem.

Mathematics Subject Classification

Notes

Acknowledgments

P. Pucci is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica “G. Severi” (INdAM) and was partially supported by the MIUR Project Aspetti variazionali e perturbativi nei problemi differenziali nonlineari, and finally the manuscript was realized within the auspices of the INDAM-GNAMPA Project 2015 titled Modelli ed equazioni non-locali di tipo frazionario (Prot_2015_000368). M. Xiang was support by the Fundamental Research Funds for the Central Universities (No. 3122015L014). B. Zhang was supported by Natural Science Foundation of Heilongjiang Province of China (No. A201306) and Research Foundation of Heilongjiang Educational Committee (No. 12541667) and Doctoral Research Foundation of Heilongjiang Institute of Technology (No. 2013BJ15).

Appendix

In this section we show that the Banach space W defined in the Introduction is a uniformly convex Banach space. We prefer to give full details for completeness, even if it could be readily seen.

Lemma 10

\(W=(W,\Vert \cdot \Vert _{W})\) is a uniformly convex Banach space.

Proof

Clearly W is complete with respect to the norm \(\Vert \cdot \Vert _{W}\). Indeed, let \(\{u_n\}_n\) be a Cauchy sequence in W. Thus, for any \(\varepsilon >0\) there exists \(\mu _\varepsilon >0\) such that if \(n,m\ge \mu _\varepsilon \), then

By the completeness of \(L^p({\mathbb {R}}^N)\), there exists \(u\in L^p({\mathbb {R}}^N)\) such that \(u_n\rightarrow u\) strongly in \(L^p({\mathbb {R}}^N)\) as \(n\rightarrow \infty \). So, there exists a subsequence \(\{u_{n_k}\}\) in W such that \(u_{n_k}\rightarrow u\) a.e. in \({\mathbb {R}}^N\) as \(k\rightarrow \infty \) (see [9, Theorem 4.9]). Therefore, by the Fatou Lemma and the second inequality in (5.1) with \(\varepsilon =1\), we have

Remark 2

By Theorem 1.21 of [1], the space W is a reflexive Banach space. With the same arguments as Lemma 10, it easily follows that \(W^{s,p}({\mathbb {R}}^N)\) is also a uniformly convex Banach space, and hence \(W^{s,p}({\mathbb {R}}^N)\) is a reflexive Banach space.